Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, x\neq 0$. $\dfrac{{t^{-3}x^{-2}}}{{(t^{3}x^{-1})^{-3}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${t^{-3}x^{-2} = t^{-3}x^{-2}}$ On the left, we have ${t^{-3}}$ to the exponent ${1}$ . Now ${-3 \times 1 = -3}$ , so ${t^{-3} = t^{-3}}$ Apply the ideas above to simplify the equation. $\dfrac{{t^{-3}x^{-2}}}{{(t^{3}x^{-1})^{-3}}} = \dfrac{{t^{-3}x^{-2}}}{{t^{-9}x^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-3}x^{-2}}}{{t^{-9}x^{3}}} = \dfrac{{t^{-3}}}{{t^{-9}}} \cdot \dfrac{{x^{-2}}}{{x^{3}}} = t^{{-3} - {(-9)}} \cdot x^{{-2} - {3}} = t^{6}x^{-5}$